Of course, and anyone else succeeding to the office of President. (That is if the Speaker of the House would be required to succeed to the office, she would need to be a Natural born citizen, else the next in line would do so in her stead.)
You are making the assumption, without much evidence that I'm aware of, that there is a difference.
From the majority opinion inWong Kim Ark:
Mr. Binney in the second edition of a paper on the Alienigenae of the United States, printed in pamphlet at Philadelphia, with a preface bearing his signature and the date of December 1, 1853, said: 'The common- law principle of allegiance was the law of all the states at the time of the Revolution and at the adoption of the constitution; and by that principle the citizens of the United States are, with the exceptions before mentioned [namely, foreign-born children of citizens, under statutes to be presently referred to], such only as are either born or made so, born within the limits and under the jurisdiction of the United States, or naturalized by the authority of law, either in one of the states before the constitution, or, since that time, by virtue of an act of the congress of the United States.' Page 20. 'The right of citizenship never descends in the legal sense, either by the common law, or under the common naturalization acts. It is incident to birth in the country, or it is given personally by statute. The child of an alien, if born in the country, is as much a citizen as the natural-born child of a citizen , and by operation of the same principle.' [169 U.S. 649, 666] Page 22, note.
This certainly implies that "citizen" and "natural born child of a citizen" are not the same thing, and in fact gives a basic definition of "Natural Born". That is "child of a citizen'"
Please ignore post 64, I was a little quick on the “post” button, and my net was slow to respond.